Structural Causal Models Revisited

Structural Causal Models (SCMs) are formalized. A thorough definition of SCMs, including their associated Directed Acyclic Graphs (DAGs), is essential for applying advanced identification strategies and estimation techniques. SCMs provide the mathematical machinery to explicitly represent causal assumptions, define interventions, and ultimately determine if causal effects are estimable from available data.

Defining Structural Causal Models

An SCM provides a complete description of how a system's variables acquire their values. Formally, a Structural Causal Model $M$ is a tuple $M = \langle U, V, F, P(U) \rangle$, where:

1. $U$ is a set of exogenous variables. These are variables determined by factors outside the model. They represent stochastic background conditions, noise, or unmodeled influences. Think of them as the fundamental sources of randomness in the system.
2. $V$ is a set of endogenous variables. These are the variables whose values are determined by other variables within the model (both endogenous and exogenous). These are typically the variables of interest whose relationships we aim to understand, such as treatments, outcomes, and confounders.
3. $F$ is a set of structural equations, one for each endogenous variable $V_i \in V$. Each equation specifies the value of $V_i$ as a function $f_i$ of other variables in the model, both endogenous ($PA_i \subseteq V \setminus \{V_i\}$) and exogenous ($U_i \subseteq U$): $$V_i = f_i(PA_i, U_i)$$ Here, $PA_i$ denotes the set of endogenous parents of $V_i$, meaning the variables in $V$ that directly affect $V_i$ according to the function $f_i$. The $U_i$ associated with $V_i$ represents the exogenous influences specific to that variable's determination, capturing any unmodeled factors or intrinsic randomness.
4. $P(U)$ is a probability distribution over the exogenous variables $U$. This distribution captures the potential correlations and dependencies among the background factors influencing the system.

Crucially, each equation $V_i = f_i(PA_i, U_i)$ represents an autonomous causal mechanism. It describes how $V_i$ is determined based on its direct causes and specific exogenous factors, independent of the mechanisms governing other variables.

Assumptions and Their Implications

The SCM definition implicitly carries significant assumptions:

• Implicit Causal Sufficiency: The standard definition assumes that for any pair of endogenous variables $V_i, V_j$, any common cause is either included in $V$ or its influence is captured by the correlations within $P(U)$. If we further assume the exogenous variables $U_i$ are mutually independent ($P(U) = \prod P(U_i)$), then the model assumes there are no unobserved common causes among the endogenous variables $V$. This strong assumption is often unrealistic in practice, and Chapter 4 addresses methods for addressing violations (unobserved confounding). If the $U_i$ are not assumed independent, their dependency structure represents latent confounding variables.
• Acyclicity (Typically): Most introductory treatments and many advanced methods assume the causal relationships encoded in $F$ do not form directed cycles. That is, a variable cannot be its own ancestor. This implies that the relationships among the endogenous variables $V$ can be represented by a Directed Acyclic Graph (DAG). We will specifically address models with cycles and feedback loops later in this chapter ("Addressing Cycles and Feedback in Causal Graphs").
• Modularity/Autonomy: This is a foundation assumption. It posits that each mechanism $f_i$ is distinct and can, in principle, be altered (e.g., by an intervention) without affecting the other mechanisms $f_j$ for $j \neq i$. This property is what allows us to precisely define and analyze the effect of interventions.

From SCMs to Graphs and Distributions

An SCM naturally induces a causal graph (often a DAG). The nodes of this graph correspond to the endogenous variables $V$. A directed edge exists from $V_j$ to $V_i$ if and only if $V_j$ is an argument in the function $f_i$ (i.e., $V_j \in PA_i$). The absence of an edge from $V_j$ to $V_i$ encodes the strong causal assumption that $V_j$ has no direct causal effect on $V_i$.

Consider a simple SCM: $U = \{U_X, U_Y, U_Z\}$ assumed independent, standard normal. $V = \{X, Y, Z\}$ $F = \{$ $X = f_X(U_X) = U_X$, $Z = f_Z(X, U_Z) = \alpha X + U_Z$, $Y = f_Y(X, Z, U_Y) = \beta X + \gamma Z + U_Y$ $\}$ $P(U)$ specifies the distributions (e.g., standard normal, independent).

This SCM induces the following graph:

The graph shows direct causal influences among endogenous variables (solid arrows) and indicates exogenous influences (dashed lines, often omitted when U are assumed independent).

An SCM $M$ defines a unique observational probability distribution $P(V)$ over the endogenous variables. This distribution arises from propagating the uncertainty from $P(U)$ through the structural equations $F$.

Interventions in SCMs: The $do$-Operator

The power of SCMs lies in their ability to formally model interventions. An intervention, denoted by the $do$-operator, represents an external action that forces a variable $X \in V$ to take on a specific value $x$, overriding its natural causal mechanism.

Formally, performing the intervention $do(X=x)$ on an SCM $M = \langle U, V, F, P(U) \rangle$ creates a modified SCM, denoted $M_x = \langle U, V, F_x, P(U) \rangle$. The only difference is in the set of functions $F_x$: the original equation for $X$, $X = f_X(PA_X, U_X)$, is replaced by the constant assignment $X=x$. All other structural equations in $F$ remain unchanged, reflecting the modularity assumption.

The interventional distribution $P(Y | do(X=x))$ (or more generally, the post-intervention distribution $P_{M_x}(V)$) is the distribution of variables resulting from this modified model $M_x$.

It is fundamental to understand that $P(Y | do(X=x))$ is and often numerically different from the observational conditional distribution $P(Y | X=x)$.

• $P(Y | X=x)$ describes the distribution of $Y$ in the subpopulation where $X$ happens to be $x$, calculated from the original SCM $M$. This reflects passive observation.
• $P(Y | do(X=x))$ describes the distribution of $Y$ when we force $X$ to be $x$, calculated from the modified SCM $M_x$. This reflects an active manipulation.

In our example graph, $P(Y | X=x)$ is influenced by the path $X \rightarrow Z \rightarrow Y$ and the direct path $X \rightarrow Y$. If $X$ and $Y$ shared an unobserved common cause represented by correlated $U_X, U_Y$, then $P(Y | X=x)$ would also reflect this non-causal correlation. However, $P(Y | do(X=x))$ reflects only the causal influence flowing out of the manipulated $X$ along the paths $X \rightarrow Y$ and $X \rightarrow Z \rightarrow Y$. It's computed in the model where $X$'s value is fixed to $x$, severing any incoming arrows to $X$.

This precise definition of interventions within the SCM framework is the basis for causal effect identification. The central question becomes: can we compute $P(Y | do(X=x))$ using only the observed distribution $P(V)$ and assumptions about the causal structure (the graph), without needing full knowledge of the functions $f_i$ or the distribution $P(U)$? The do-calculus, explored next, provides formal rules for answering this question. Mastering the SCM formalism is the necessary first step towards applying these powerful tools.